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(CNN)It's the most coveted legal post in Washington, but few people outside of the Beltway know it even exists.

And now the plum job -- that of solicitor general -- is up for grabs.

So far, two names have emerged as strong potentials, but Donald Trump has yet to make the official pick.

The two possible nominees reflect the opposite ends of a spectrum. The first, Charles J. Cooper, is a fixture of the Washington legal scene. The second, George Conway, would be an outside-of-the-box choice. Others are jockeying for the job and the situation is fluid, according to sources close to the transition.

Anticipation -- in the corridors of elite law firms -- is mounting.

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The solicitor general is the person who argues on behalf of the government in the Supreme Court of the United States. That means the "S.G." stands in the ornate chamber, faces nine of the top legal minds in the country, and gets constantly interrupted with a deluge of questions from a very hot bench.

Past solicitors general relish the experience.

In fact, the legendary Thurgood Marshall, who changed the law as an NAACP-LDF attorney and later served on the Supreme Court, once said that his time as solicitor general from 1965-1967 was "the best job I've ever had, bar none."

Marshall made the revelation to Lincoln Caplan who published a book in 1987 called "The Tenth Justice: The Solicitor General and the Rule of Law."

Caplan said the job attracts those "endowed with a particular kind of logical mind, a capacious memory, but also a passion for the stuff of Supreme Court practice which is opinions and more opinions."

While the solicitor general reports to the attorney general, he or she has responsibilities with other branches of government as well. That brings with it something rare in a government job: a large degree of independence.

But the "inestimable privilege" of representing the government is only the "tip of the iceberg" said Paul D. Clement who served as solicitor general during the George W. Bush administration.

"What really makes the solicitor general's job unique -- and makes it such an incomparable experience -- is the behind-the-scenes role of determining whether the United States will appeal‎ in thousands of cases each year and shaping the positions taken in those cases," he said.

"The solicitor general probably makes more strategic calls about whether and how to appeal in a year than any other lawyer makes in a life time," he said.

Conway or Cooper?

Sources say that Trump is considering Conway, a corporate attorney out of New York who is married to Kellyanne Conway a senior adviser to the President-elect. In many ways, picking Conway -- a partner of Wachtell, Lipton, Rosen & Katz -- would be a vintage move for Trump because Conway is an outside-the-Beltway choice. Although Conway has only argued one case before the court, it was a big win. He's got all of the qualifications for the job.

The fact that he only argued once gives him more experience than Elena Kagan who served as solicitor general from 2009 to 2010. She had never argued before the justices.

Another person on Trump's list is Cooper.

Unlike Conway, Cooper is often at the Supreme Court. He hails from Alabama, clerked for Chief Justice William Rehnquist, and has argued several cases before the Supreme Court. It was Cooper who stepped in to defend California's Proposition 8, a referendum that banned gay marriage. Most recently he worked behind the scenes to help prepare Sen. Jeff Sessions for his confirmation hearing for attorney general.

Why would an established lawyer give up a law practice to go back into government? Because the role is considered a dream job. Caplan says the job means a lawyer gets to "practice at the apex of the American constitutional system" and run a staff of about 20 exceedingly smart lawyers. "They follow the court more closely than anyone in the country including the best constitutional scholars," Caplan said, adding they are "people who love the law in a particular way."

Seth Waxman, who served from 1997 to 2001 put it this way: "In a country of 280 million souls, how does one ascertain what the interests of the United States are in litigation?" he queried at a 2002 conference hosted by Brigham Young University. "That is the solicitor general's most challenging and exciting mission."

It was only in 1870 that a law was passed to establish the office and today there are more than 20 lawyers in the office.

To some, it is a step that leads to the Supreme Court -- indeed three current justices served in the office.

At his confirmation hearing Chief Justice John Roberts reflected on his years serving in the office, "I always found it very moving," he told the senators, "to stand before the justices and say, 'I speak for my country.'"